Question

The number of hits to a website follows a Poisson process. Hits occur at the rate of 2.9 per minute between​ 7:00 P.M. and

11​:00 P.M. Given below are three scenarios for the number of hits to the website. Compute the probability of each scenario between 8 : 44 P.M.

and 8​:46 P.M. Interpret each result.

​(a) exactly six

​(b) fewer than six

​(c) at least six

Answer 1

The number of hits on website follows Poisson process: It's probability is:

P(x) = e-λt*(λt)x/x!

Hits occurrence rate = 2.9 per min

And random variable x is defined over the interval between 8:44 PM and 8​:46 PM, so t = 2 minutes

Answer a)

Probability of exactly six

P(6) = e-2.9*2*(2.9*2)6/6!

P(6) = 0.1601

Answer b)

Probability of fewer than six

P(x < 6) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5)

P(0) = e-2.9*2*(2.9*2)0/0! = 0.0030

P(1) = e-2.9*2*(2.9*2)1/1! = 0.0176

P(2) = e-2.9*2*(2.9*2)2/2! = 0.0509

P(3) = e-2.9*2*(2.9*2)3/3! = 0.0985

P(4) = e-2.9*2*(2.9*2)4/4! = 0.1428

P(5) = e-2.9*2*(2.9*2)5/5! = 0.1656

P(x < 6) = 0.0030 + 0.0176 + 0.0509 + 0.0985 + 0.1428 + 0.1656

P(x < 6) = 0.4784

Answer c)

Probability of at least six

P(x ≥ 6) = 1 - P(x < 6) = 1 - 0.4784

P(x ≥ 6) = 0.5216